Offering a Draw or Accepting a Draw

Can you only offer a draw on your own turn ? If you offer a draw on your own turn, does your opponent have to claim it immediately for the draw to stand or can he play a move and then claim the draw later ? I would be grateful for any clarification.

Yes and yes.  Note that your opponent doesn't formally have to reject or even acknowledge your draw offer.  Making his next move rejects the offer.

Of course in casual OTB games no one cares.  "Hey, is that draw offer still good?"   "Yeah sure, this isn't going anywhere"  etc.

Thanks. I am assuming that what you say applies to correspondence games on chess.com as well as live games ?

As far as I know.  There may be some minor differences in the way draws are claimed though.  I think for 3 fold repetition and 50 move the offer draw button changes to claim draw in correspondence and it doesn't in live, but I really don't remember for sure... 

if your opponent could claim a draw after a move, then if your opponent starts losing after the draw has been offered, they could accept it then,